A project will take 63 days by team A, and then another 28 days by team B to complete. If both teams are to work on this project together, it will take 48 days to complete. If team A works for 42 days, how many days will it take team B alone to complete the remaining part of the project?

Let's break down the problem and solve it step by step.

### Step 1: Determine work rates of teams A and B
- **Team A** can complete the project alone in 63 days. Therefore, team A's rate of work is:
  \[
  \text{Team A's work rate} = \frac{1}{63} \text{ of the project per day.}
  \]
- **Team B** can complete the project alone in 28 days. Therefore, team B's rate of work is:
  \[
  \text{Team B's work rate} = \frac{1}{28} \text{ of the project per day.}
  \]

### Step 2: Combined work rate of both teams
When both teams work together, they can complete the project in 48 days. The combined work rate is:
\[
\frac{1}{A} + \frac{1}{B} = \frac{1}{48}
\]
We already know the individual rates of teams A and B, so:
\[
\frac{1}{63} + \frac{1}{28} = \frac{1}{48}
\]
To solve this, first find a common denominator for \(\frac{1}{63}\) and \(\frac{1}{28}\). The least common denominator of 63 and 28 is 252, so:
\[
\frac{1}{63} = \frac{4}{252}, \quad \frac{1}{28} = \frac{9}{252}
\]
Thus,
\[
\frac{1}{63} + \frac{1}{28} = \frac{4}{252} + \frac{9}{252} = \frac{13}{252} = \frac{1}{48}
\]
This confirms that their combined work rate is indeed \(\frac{1}{48}\) of the project per day.

### Step 3: Team A works for 42 days
If team A works alone for 42 days, the amount of work it completes is:
\[
\text{Work done by Team A in 42 days} = 42 \times \frac{1}{63} = \frac{42}{63} = \frac{2}{3} \text{ of the project.}
\]
So, after 42 days, \(\frac{2}{3}\) of the project is completed by team A, leaving:
\[
1 - \frac{2}{3} = \frac{1}{3} \text{ of the project remaining.}
\]

### Step 4: Determine how long it will take team B to finish the remaining work
Team B needs to complete the remaining \(\frac{1}{3}\) of the project. Since team B's work rate is \(\frac{1}{28}\) of the project per day, the number of days \(d\) it will take team B to complete \(\frac{1}{3}\) of the project is:
\[
\frac{1}{28} \times d = \frac{1}{3}
\]
Solve for \(d\):
\[
d = \frac{1}{3} \times 28 = \frac{28}{3} = 9.33 \text{ days.}
\]

Thus, it will take team B approximately **9.33 days** (or 9 days and 8 hours) to complete the remaining part of the project.

### Summary:
After team A works for 42 days, team B will take about **9.33 days** to finish the remaining part of the project.

Do you want details or have any questions?

Here are some related questions to explore further:
1. What would happen if team A worked for only 30 days instead of 42?
2. If team B starts working with team A on day 20, how long would it take to complete the project?
3. How would the project time change if a third team (team C) joined the project and could complete it in 40 days?
4. If team A's efficiency drops by 10%, how does that affect the total time needed for the project?
5. What would the total project time be if both teams work part-time, completing 70% of their normal rate?

**Tip:** When multiple teams work together on a task, finding their combined rate by adding individual work rates is a quick way to estimate project completion times.

Learn how to solve a work rate problem involving teams A and B, where team A works for 42 days, and then team B completes the remaining work. This problem uses key algebraic concepts such as work rate and combined rates. Suitable for students in Grades 9-11.

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